To solve the inequality [tex]\( 2x - 6 \geq 6(x - 2) + 8 \)[/tex], we will follow a series of algebraic steps to isolate [tex]\( x \)[/tex]. Let's work through these steps carefully:
1. Distribute the 6 on the right side of the inequality:
[tex]\[
2x - 6 \geq 6(x - 2) + 8
\][/tex]
[tex]\[
2x - 6 \geq 6x - 12 + 8
\][/tex]
2. Simplify the right side:
[tex]\[
2x - 6 \geq 6x - 4
\][/tex]
3. Move all [tex]\( x \)[/tex]-terms to one side by subtracting [tex]\( 6x \)[/tex] from both sides:
[tex]\[
2x - 6 - 6x \geq 6x - 4 - 6x
\][/tex]
[tex]\[
-4x - 6 \geq -4
\][/tex]
4. Isolate [tex]\( x \)[/tex] by adding 6 to both sides:
[tex]\[
-4x - 6 + 6 \geq -4 + 6
\][/tex]
[tex]\[
-4x \geq 2
\][/tex]
5. Divide both sides by -4 and reverse the inequality sign (since dividing by a negative number reverses the inequality sign):
[tex]\[
x \leq \frac{2}{-4}
\][/tex]
[tex]\[
x \leq -\frac{1}{2}
\][/tex]
The solution set for the inequality is [tex]\( x \leq -\frac{1}{2} \)[/tex].
To represent this on a number line:
- Draw a number line.
- Locate [tex]\(-\frac{1}{2}\)[/tex] on the number line.
- Draw a closed circle (or dot) at [tex]\(-\frac{1}{2}\)[/tex] to show that [tex]\(-\frac{1}{2}\)[/tex] is included in the solution (since the inequality is "less than or equal to").
- Shade the number line to the left of [tex]\(-\frac{1}{2}\)[/tex] to indicate all values less than [tex]\(-\frac{1}{2}\)[/tex] are part of the solution.
Here is a visual representation of what the number line should look like:
[tex]\[
\begin{array}{cccccccccccc}
& & & & & & \bullet & \longleftarrow & & & & & \\
& & & & & -1 & & & & & 0 & & \\
\end{array}
\][/tex]
The [tex]\(\bullet\)[/tex] at [tex]\(-\frac{1}{2}\)[/tex] represents [tex]\(-\frac{1}{2}\)[/tex] being included in the solution set, and the arrow to the left shows all numbers less than or equal to [tex]\(-\frac{1}{2}\)[/tex].
Thus, the number line correctly represents the solution set for the inequality [tex]\( 2x - 6 \geq 6(x - 2) + 8 \)[/tex].
